New Theory Explains Speciation in Diffusion Models

Generative diffusion models exhibit a phenomenon called "speciation," where initially undifferentiated trajectories commit to distinct data classes during the denoising process. Existing theories often simplify this by assuming large-dimensional spaces or symmetric bifurcations. This research proposes a more intrinsic theory of speciation, specifically for diffusion models operating on compact Riemannian manifolds, which explicitly accounts for the underlying geometry. The theory characterizes speciation by analyzing bifurcations of the critical points of the evolving probability density. It uses a spectral heat-kernel representation to highlight the manifold's geometric influence, while Poincaré-Hopf and Morse theory impose global constraints on the number and types of score equilibria. For mixtures of heat kernels, the paper proves that generic speciation events involve a one-dimensional critical kernel and follow an A2 fold normal form, with pitchforks arising from specific symmetric configurations. Geometry-dependent estimates for speciation times are derived, and the structural stability of nondegenerate folds under score perturbations is established. The theory is illustrated on a sphere using von Mises-Fisher distributions, demonstrating various bifurcation types and hierarchical speciations. This work provides a deeper, geometrically informed understanding of how diffusion models generate diverse data.